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arXiv:2304.05286 · Physical Review Letters

Unveiling the Non-Abelian Statistics of D ( S 3 ) Anyons Using a Classical Photonic Simulator

Suraj Goel, Matthew Reynolds, Matthew Girling, Will McCutcheon, Saroch Leedumrongwatthanakun, Vatshal Srivastav, David Jennings, Mehul Malik, Jiannis K. Pachos

Published Mar 13, 2024• 8 citations
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Abstract

Simulators can realize novel phenomena by separating them from the complexities of a full physical implementation. Here, we put forward a scheme that can simulate the exotic statistics of D(S_{3}) non-Abelian anyons with minimal resources. The qudit lattice representation of this planar code supports local encoding of D(S_{3}) anyons. As a proof-of-principle demonstration, we employ a classical photonic simulator to encode a single qutrit and manipulate it to perform the fusion and braiding properties of non-Abelian D(S_{3}) anyons. The photonic technology allows us to perform the required nonunitary operations with much higher fidelity than what can be achieved with current quantum computers. Our approach can be directly generalized to larger systems or to different anyonic models, thus enabling advances in the exploration of quantum error correction and fundamental physics alike.

DOI
10.1103/physrevlett.132.110601

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Simulators can realize novel phenomena by separating them from the complexities of a full physical implementation. Here, we put forward a scheme that can simulate the exotic statistics of D(S_{3}) non-Abelian anyons with minimal resources. The qudit lattice representation of this planar code supports local encoding of D(S_{3}) anyons. As a proof-of-principle demonstration, we employ a classical photonic simulator to encode a single qutrit and manipulate it to perform the fusion and braiding properties of non-Abelian D(S_{3}) anyons. The photonic technology allows us to perform the required nonunitary operations with much higher fidelity than what can be achieved with current quantum computers. Our approach can be directly generalized to larger systems or to different anyonic models, thus enabling advances in the exploration of quantum error correction and fundamental physics alike.
Abstract We consider a lattice of d = 6 qudits that supports D ( S 3 ) non-Abelian anyons. We present a method for implementing both braiding and fusion evolutions using only the operators that create and measure anyons, without requiring additional dynamical control. This provides a minimal protocol demonstrating that D ( S 3 ) anyons can generate magic states, thereby establishing their universality for quantum computation. Furthermore, we show that the entire scheme can be encoded in just two qudits, offering a compact blueprint that is inherently scalable and readily implementable in current quantum platforms.
Demonstration of magic state power of D(S3)\mathbf{D}(\mathbf{S}_{3})D(S3​) anyons with two qudits
We consider a lattice of d=6d=6d=6 qudits that supports D(S3)\mathbf{D}(\mathbf{S}_3)D(S3​) non-Abelian anyons. We present a method for implementing both braiding and fusion evolutions using only the operators that create and measure anyons, without requiring additional dynamical control. This provides a minimal protocol demonstrating that D(S3)\mathbf{D}(\mathbf{S}_3)D(S3​) anyons can generate magic states, thereby establishing their universality for quantum computation. Furthermore, we show that the entire scheme can be encoded in just two qudits, offering a compact blueprint that is inherently scalable and readily implementable in current quantum platforms.
Non-Abelian anyons offer a route to fault-tolerant and universal quantum computing, but experimental access to their defining braiding and fusion data remains limited by the resource overhead of implementing extended anyonic processes on quantum hardware. Here we introduce and experimentally realise a measurement-only protocol based on temporally ordered ribbon operations that reconstructs the non-Abelian braiding and fusion primitives of the quantum double model D(S3)D(S_3)D(S3​) without physical anyon transport. We implement a reduced two-qutrit version of the protocol on Quantinuum's H2 trapped-ion processors, realising ancilla-assisted ribbon operations and anyonic charge projections in a qubit encoding. We reconstruct the squared braiding phases and fusion amplitudes using an adapted Hadamard test and post-selected measurements, respectively. The associated braiding and fusion transformations reproduce their ideal actions with average normalised output-state fidelities of F‾R=0.9988\overline{\mathcal{F}}_{R}=0.9988FR​=0.9988 and F‾F=0.9987\overline{\mathcal{F}}_{F}=0.9987FF​=0.9987. Combining these primitives produces a non-Clifford braid and a non-stabilizer resource state, supporting measurement-only anyonic encodings as building blocks for larger topologically encoded quantum processors.
Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons, i.e., the Ising anyons. In particular, we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type, and can be explored to encode topological logical qubits. Furthermore, we demonstrate how to implement both single- and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits. Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons, offering a new lens into the study of such peculiar quasiparticles.
Anyons are exotic quasiparticles living in two dimensions that do not fit into the usual categories of fermions and bosons, but obey a new form of fractional statistics. Following a recent proposal [Phys. Rev. Lett. 98, 150404 (2007)], we present an experimental demonstration of the fractional statistics of anyons in the Kitaev spin lattice model using a photonic quantum simulator. We dynamically create the ground state and excited states (which are six-qubit graph states) of the Kitaev model Hamiltonian, and implement the anyonic braiding and fusion operations by single-qubit rotations. A phase shift of pi related to the anyon braiding is observed, confirming the prediction of the fractional statistics of Abelian 1/2 anyons.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.